A304864 -id:A304864

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A304319

O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * x ) / A(x) = 0 for n>0.

+10
7

1, 2, 10, 104, 1772, 42408, 1303504, 48736000, 2139552016, 107629121888, 6094743943584, 383305860004992, 26491391713168640, 1994924925169038848, 162537118868301414912, 14243360542620058589184, 1335710880923054761115904, 133461369304858515494530560, 14154134380237986764584033792, 1587931951984022880659170662400

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

It is striking that the coefficients of o.g.f. A(x) consist entirely of integers.

Note that if [x^n] exp( (n+1)*(n+2)*x ) / G(x) = 0 then G(x) does not consist entirely of integer coefficients.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..400

FORMULA

a(n) ~ sqrt(1-c) * 2^(2*n) * n^(n - 1/2) / (sqrt(Pi) * c^(n + 1/2) * (2-c)^n * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - Vaclav Kotesovec, Aug 31 2020

EXAMPLE

O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 104*x^3 + 1772*x^4 + 42408*x^5 + 1303504*x^6 + 48736000*x^7 + 2139552016*x^8 + 107629121888*x^9 + 6094743943584*x^10 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp(n*(n+1)*x) / A(x) begins:

n=0: [1, -2, -12, -432, -32640, -4176000, -804504960, -216834831360, ...];

n=1: [1, 0, -16, -520, -36432, -4520768, -856647680, -228458074752, ...];

n=2: [1, 4, 0, -648, -46032, -5341824, -974612736, -254049782400, ...];

n=3: [1, 10, 84, 0, -56832, -6922368, -1194341760, -299397745152, ...];

n=4: [1, 18, 308, 4448, 0, -8528000, -1573784960, -376524725760, ...];

n=5: [1, 28, 768, 20088, 444720, 0, -1938504960, -502258872960, ...];

n=6: [1, 40, 1584, 61560, 2286768, 72032832, 0, -618983309952, ...];

n=7: [1, 54, 2900, 154352, 8074368, 404450176, 17201640064, 0, ...];

n=8: [1, 70, 4884, 339120, 23357568, 1583068032, 102886277760, 5682964174848, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+1)*x ) / A(x) = 0 for n>=0.

LOGARITHMIC DERIVATIVE.

The logarithmic derivative of A(x) yields the o.g.f. of A304317:

A'(x)/A(x) = 2 + 16*x + 260*x^2 + 6200*x^3 + 191832*x^4 + 7235152*x^5 + 320372320*x^6 + 16243028896*x^7 + 926219213216*x^8 + 58608051937536*x^9 + 4072302306624576*x^10 + ...+ A304317(n)*x^n + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m*(m-1) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}

for(n=0, 25, print1( a(n), ", "))

CROSSREFS

Cf. A304317, A304318, A304320, A304863, A304864, A304865.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 11 2018

STATUS

approved

A304863

O.g.f. A(x) satisfies: [x^n] exp( n*(n+2) * x ) / A(x) = 0 for n>0.

+10
3

1, 3, 17, 180, 3079, 74271, 2308940, 87438684, 3888682559, 198073751505, 11348409001233, 721483807171188, 50361931297722244, 3827114191186713588, 314413091556481490640, 27761835704580647457012, 2621495363274661266785679, 263593068966612639018287637, 28117066903131481643928647363, 3171150259810035292799245555884, 377044852592342586608552585592079

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..20.

EXAMPLE

O.g.f.: A(x) = 1 + 3*x + 17*x^2 + 180*x^3 + 3079*x^4 + 74271*x^5 + 2308940*x^6 + 87438684*x^7 + 3888682559*x^8 + 198073751505*x^9 + 11348409001233*x^10 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp(n*(n+2)*x) / A(x) begins:

n=0: [1, -3, -16, -630, -50112, -6665400, -1325134080, -366674636160, ...];

n=1: [1, 0, -25, -828, -58779, -7479072, -1452239685, -395811449100, ...];

n=2: [1, 5, 0, -1078, -78464, -9183672, -1700942720, -450843184000, ...];

n=3: [1, 12, 119, 0, -99387, -12381300, -2151101205, -544666984560, ...];

n=4: [1, 21, 416, 6858, 0, -15533496, -2923952256, -703585823616, ...];

n=5: [1, 32, 999, 29540, 730213, 0, -3653179205, -962999633260, ...];

n=6: [1, 45, 2000, 86922, 3589056, 124275528, 0, -1200826684800, ...];

n=7: [1, 60, 3575, 210672, 12162501, 668679228, 30900268395, 0, ...];

n=8: [1, 77, 5904, 449930, 33949888, 2513449800, 177544721920, 10559736679040, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n+2)*x ) / A(x) = 0 for n>=0.

RELATED SERIES.

The logarithmic derivative of A(x) yields:

A'(x)/A(x) = 3 + 25*x + 414*x^2 + 10109*x^3 + 320253*x^4 + 12346720*x^5 + 557708406*x^6 + 28786882117*x^7 + 1668054884229*x^8 + 107077380781005*x^9 + ...

1 - 1/A(x) = 3*x + 8*x^2 + 105*x^3 + 2088*x^4 + 55545*x^5 + 1840464*x^6 + 72752904*x^7 + 3334122880*x^8 + 173569203225*x^9 + 10108800765000*x^10 + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m+1) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}

for(n=0, 25, print1( a(n), ", "))

CROSSREFS

Cf. A304322, A304318, A304319, A304864, A304865.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 19 2018

STATUS

approved

A304865

O.g.f. A(x) satisfies: [x^n] exp( n*(n+4) * x ) / A(x) = 0 for n>0.

+10
3

1, 5, 37, 434, 7751, 193101, 6200872, 242667316, 11144759839, 585318934391, 34511211188637, 2253285826341378, 161201686356627524, 12530568505972885004, 1051099249634285619168, 94603882448795669308980, 9092091650779263675187695, 929177036869575506758681035, 100608724821944458615599713935, 11504982932704269804549116593702, 1385525417578463389730054278506959

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..20.

EXAMPLE

O.g.f.: A(x) = 1 + 5*x + 37*x^2 + 434*x^3 + 7751*x^4 + 193101*x^5 + 6200872*x^6 + 242667316*x^7 + 11144759839*x^8 + 585318934391*x^9 + 34511211188637*x^10 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp(n*(n+4)*x) / A(x) begins:

n=0: [1, -5, -24, -1134, -100608, -14542200, -3095496000, -907608905280, ...];

n=1: [1, 0, -49, -1744, -128763, -17383400, -3572628125, -1024052930280, ...];

n=2: [1, 7, 0, -2430, -189600, -22895928, -4410982656, -1218708054720, ...];

n=3: [1, 16, 207, 0, -250107, -33107544, -5910144669, -1540910769048, ...];

n=4: [1, 27, 680, 13970, 0, -42775928, -8486494016, -2090851421760, ...];

n=5: [1, 40, 1551, 56376, 1681797, 0, -10852876125, -2994692165280, ...];

n=6: [1, 55, 2976, 156546, 7748832, 316211400, 0, -3807596825280, ...];

n=7: [1, 72, 5135, 360920, 24718725, 1597879072, 85448027299, 0, ...];

n=8: [1, 91, 8232, 738450, 65376768, 5650680456, 462123838656, 31350065660352, 0, ...]; ...

RELATED SERIES.

The logarithmic derivative of A(x) yields:

A'(x)/A(x) = 5 + 49*x + 872*x^2 + 22661*x^3 + 759915*x^4 + 30843448*x^5 + 1459277062*x^6 + 78529473925*x^7 + 4724556111179*x^8 + 313739794874469*x^9 + ...

1 - 1/A(x) = 5*x + 12*x^2 + 189*x^3 + 4192*x^4 + 121185*x^5 + 4299300*x^6 + 180081132*x^7 + 8675950464*x^8 + 471853727865*x^9 + 28563862383700*x^10 + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m+3) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}

for(n=0, 25, print1( a(n), ", "))

CROSSREFS

Cf. A304322, A304318, A304319, A304863, A304864.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 19 2018

STATUS

approved

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